Answer:
The correct option is: 58.2 years.
Step-by-step explanation:
The half-life formula is: [tex]N(t)= N_{0}(\frac{1}{2})^\frac{t}{t_{1/2}}[/tex] , where [tex]N_{0}=[/tex] Original amount, [tex]N(t)=[/tex] Final amount after [tex]t[/tex] years and [tex]t_{1/2}=[/tex] Half-life in years.
The half life of Pb-210 is 22 years. So, [tex]t_{1/2}= 22[/tex] years.
A decayed animal shows 16% of the original Pb-210 remains. That means, if [tex]N_{0}=100[/tex], then [tex]N(t)= 16[/tex].
Plugging these values into the above formula, we will get......
[tex]16= 100(\frac{1}{2})^\frac{t}{22}\\ \\ \frac{16}{100}= \frac{100(\frac{1}{2})^\frac{t}{22}}{100}\\ \\ 0.16=(\frac{1}{2})^\frac{t}{22}[/tex]
Taking logarithm on both sides.......
[tex]log(0.16)=log[(\frac{1}{2})^\frac{t}{22}]\\ \\ log(0.16)=\frac{t}{22}log(\frac{1}{2})\\ \\ \frac{t}{22}=\frac{log(0.16)}{log(\frac{1}{2})}\\ \\ t= 22*\frac{log(0.16)}{log(\frac{1}{2})}=58.1648... \approx 58.2[/tex]
(Rounded to the nearest tenth)
So, the animal has been deceased for 58.2 years.
